In the classic online min-cost matching problem, the goal is to match a sequence of requests that arrive dynamically over time to a set of static servers, aiming to minimize the total cost of the matching. This assumes that there are two distinct "sides" and that only one of these sides arrives online, but many of the motivating applications violate these assumptions. We study online min-cost perfect-matching when all participants arrive online and, upon arrival, they need to either be matched to someone from a waiting pool or to join the waiting pool. We evaluate the competitive ratios achievable in different input models and show that for both the adversarial and the random-order input models the competitive ratio of any algorithm is unbounded. In contrast, for i.i.d. arrivals we give a O( log^2n)-competitive algorithm, even if the distribution that generates these arrivals is unknown to the algorithm. This result implies a rare example of separation in the achievable competitive ratio between the random-order and the unknown-i.i.d. input models.

Robust optimization protects against uncertainty by optimizing for the worst case over a prescribed uncertainty set. This protection can be overly conservative when forecasts, historical data, or learned predictions indicate a more likely scenario. We introduce a framework for robust optimization with predictions. The input consists of an uncertainty set together with a distinguished predicted scenario, and the goal is to compute a single solution that is both consistent, meaning near-optimal for the predicted scenario, and robust, meaning competitive with the classical min-max robust optimum. Unlike in standard learning-augmented algorithms, the prediction does not merely estimate the realized input; it creates a separate benchmark, the predicted optimum, which must be balanced against the min-max robust optimum. We study this framework for makespan scheduling with uncertain processing times and give a structural classification across standard uncertainty models and machine environments. For interval uncertainty, we obtain a smooth (1+1/λ,1+λ) consistency-robustness tradeoff for restricted-assignment and related machines. Furthermore, we prove that unrelated machines admit no constant tradeoff. For budgeted uncertainty, we obtain a (1+1/λ,2+λ) tradeoff for restricted assignment. Our analysis is based on a duality-based reduction to an interval-like upper envelope. We complement this with a lower bound showing that related machines admit no constant tradeoff even when only one job may deviate. For arbitrary uncertainty sets, we obtain constant tradeoffs for identical machines via a support-function block construction, and prove impossibility for restricted assignment. Our results show that the possibility of combining consistency and robustness in robust scheduling depends critically on the interaction between the uncertainty model and the machine environment.

For the classical maximum coverage problem, the greedy algorithm achieves a worst-case 1-1/e approximation, which is optimal unless P = NP. The notion of coverage appears in a wide range of optimization tasks, where empirical evaluations indicate approximation ratios close to 1 for the greedy algorithm on real data. Random models have provided average-case justifications for the empirical performance of many well-known algorithms, but little is known about the average-case performance of greedy for maximum coverage. We analyze the expected approximation ratio of the greedy algorithm in a random model, which we call the left-regular random model. We first show that, for all parameter settings of this model, the expected approximation ratio of the greedy algorithm improves by a constant over its worst-case 1-1/e guarantee. We then identify two simple conditions, either of which ensures that the expected approximation ratio is close to 1 for sufficiently large graphs. Finally, we show that there is a regime where greedy does not achieve an expected approximation better than 0.94. To obtain these results, we develop analytical tools, including a novel application of the differential equation method and a connection to maximum matching in Erdős-Rényi graphs, which may be of independent interest for other random models.

The knapsack secretary problem is a generalization of the classical secretary problem where the accepted items must satisfy a knapsack constraint. A line of work has developed constant-competitive algorithms for this problem, with successive improvements culminating in the current best-known competitive ratio of $0.153$. A natural open question was whether the optimal $1/e$ competitive ratio for the classical secretary problem is also achievable for the knapsack secretary problem. We answer this question negatively by showing that no $(1/e - 0.0001)$-competitive algorithm exists for the knapsack secretary problem. The analysis of the family of hard instances we construct proceeds in three steps. First, we reduce the cardinal problem on these instances to an almost-ordinal problem. Second, we formulate a linear program that captures the performance of almost-ordinal algorithms on this instance family. Finally, we exhibit a feasible dual solution whose objective value is strictly below $1/e$. We also give an algorithm that improves the best-known competitive ratio from $0.153$ to $0.178$.

We revisit the well-studied problem of budget-feasible procurement, where a buyer with a strict budget constraint seeks to acquire services from a group of strategic providers (the sellers). During the last decade, several strategyproof budget-feasible procurement auctions have been proposed, aiming to maximize the value of the buyer, while eliciting each seller's true cost for providing their service. These solutions predominantly take the form of randomized sealed-bid auctions: they ask the sellers to report their private costs and then use randomization to determine which subset of services will be procured and how much each of the chosen providers will be paid, ensuring that the total payment does not exceed budget. Our main result in this paper is a novel method for designing budget-feasible auctions, leading to solutions that outperform the previously proposed auctions in multiple ways. First, our solutions take the form of descending clock auctions, and thus satisfy a list of properties, such as obvious strategyproofness, group strategyproofness, transparency, and unconditional winner privacy; this makes these auctions much more likely to be used in practice. Second, in contrast to previous results that heavily depend on randomization, our auctions are deterministic. As a result, we provide an affirmative answer to one of the main open questions in this literature, asking whether a deterministic strategyproof auction can achieve a constant approximation when the buyer's valuation function is submodular over the set of services. In addition, we also provide the first deterministic budget-feasible auction that matches the approximation bound of the best-known randomized auction for the class of subadditive valuations. Finally, using our method, we improve the best-known approximation factor for monotone submodular valuations, which has been the focus of most of the prior work.

An important challenge in robust machine learning is when training data is provided by strategic sources who may intentionally report erroneous data for their own benefit. A line of work at the intersection of machine learning and mechanism design aims to deter strategic agents from reporting erroneous training data by designing learning algorithms that are strategyproof. Strategyproofness is a strong and desirable property, but it comes at a cost in the approximation ratio of even simple risk minimization problems. In this paper, we study strategyproof regression and classification problems in a model with advice. This model is part of a recent line on mechanism design with advice where the goal is to achieve both an improved approximation ratio when the advice is correct (consistency) and a bounded approximation ratio when the advice is incorrect (robustness). We provide the first non-trivial consistency-robustness tradeoffs for strategyproof regression and classification, which hold for simple yet interesting classes of functions. For classes of constant functions, we give a deterministic and strategyproof mechanism that is, for any γ∈ (0, 2], 1+γ consistent and 1 + 4/γ robust and provide a lower bound that shows that this tradeoff is optimal. We extend this mechanism and its guarantees to homogeneous linear regression over ℝ. In the binary classification problem of selecting from three or more labelings, we present strong impossibility results for both deterministic and randomized mechanism. Finally, we provide deterministic and randomized mechanisms for selecting from two labelings.

We study the problem of designing procurement auctions for the strategic uncapacitated facility location problem: a company needs to procure a set of facility locations in order to serve its customers and each facility location is owned by a strategic agent. Each owner has a private cost for providing access to their facility (e.g., renting it or selling it to the company) and needs to be compensated accordingly. The goal is to design truthful auctions that decide which facilities the company should procure and how much to pay the corresponding owners, aiming to minimize the total cost, i.e., the monetary cost paid to the owners and the connection cost suffered by the customers (their distance to the nearest facility). We evaluate the performance of these auctions using the \emph{frugality ratio}. We first analyze the performance of the classic VCG auction in this context and prove that its frugality ratio is exactly $3$. We then leverage the learning-augmented framework and design auctions that are augmented with predictions regarding the owners' private costs. Specifically, we propose a family of learning-augmented auctions that achieve significant payment reductions when the predictions are accurate, leading to much better frugality ratios. At the same time, we demonstrate that these auctions remain robust even if the predictions are arbitrarily inaccurate, and maintain reasonable frugality ratios even under adversarially chosen predictions. We finally provide a family of ``error-tolerant'' auctions that maintain improved frugality ratios even if the predictions are only approximately accurate, and we provide upper bounds on their frugality ratio as a function of the prediction error.

In the strategic facility location problem, a set of agents report their locations in a metric space and the goal is to use these reports to open a new facility, minimizing an aggregate distance measure from the agents to the facility. However, agents are strategic and may misreport their locations to influence the facility’s placement in their favor. The aim is to design truthful mechanisms, ensuring agents cannot gain by misreporting. This problem was recently revisited through the learning-augmented framework, aiming to move beyond worst-case analysis and design truthful mechanisms that are augmented with (machine-learned) predictions. The focus of this work was on mechanisms that are deterministic and augmented with a prediction regarding the optimal facility location. In this paper, we provide a deeper understanding of this problem by exploring the power of randomization as well as the impact of different types of predictions on the performance of truthful learning-augmented mechanisms. We study both the single-dimensional and the Euclidean case and provide upper and lower bounds regarding the achievable approximation of the optimal egalitarian social cost.

Aiming to overcome some of the limitations of worst-case analysis, the recently proposed framework of "algorithms with predictions" allows algorithms to be augmented with a (possibly erroneous) machine-learned prediction that they can use as a guide. In this framework, the goal is to obtain improved guarantees when the prediction is correct, which is called \emph{consistency}, while simultaneously guaranteeing some worst-case bounds even when the prediction is arbitrarily wrong, which is called \emph{robustness}. The vast majority of the work on this framework has focused on a refined analysis of online algorithms augmented with predictions regarding the future input. A subsequent line of work has also successfully adapted this framework to mechanism design, where the prediction is regarding the private information of strategic agents. In this paper, we initiate the study of online mechanism design with predictions, which combines the challenges of online algorithms with predictions and mechanism design with predictions. We consider the well-studied problem of designing a revenue-maximizing auction to sell a single item to strategic bidders who arrive and depart over time, each with an unknown, private, value for the item. We study the learning-augmented version of this problem where the auction designer is given a prediction regarding the maximum value over all agents. Our main result is a strategyproof mechanism whose revenue guarantees are $\alpha$-consistent with respect to the highest value and $(1-\alpha^2)/4$-robust with respect to the second-highest value, for $\alpha \in [0,1]$. We show that this tradeoff is optimal within a broad and natural family of auctions, meaning that any $\alpha$-consistent mechanism in that family has robustness at most $(1-\alpha^2)/4$. Finally, we extend our mechanism to also achieve expected revenues proportional to the prediction quality.

The assignment game, introduced by Shapley and Shubik [1971], is a classic model for two-sided matching markets between buyers and sellers. In the original assignment game, it is assumed that payments lead to transferable utility and that buyers have unit-demand valuations for the items being sold. There has since been substantial work studying various extensions of the assignment game. The first main area of extension is to imperfectly transferable utility, which is when frictions, taxes, or fees impede the transfer of money between agents. The second is with more complex valuation functions, in particular gross substitutes valuations, which describe substitutable goods. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in each of these two settings. However, these lines of work have been mostly independent, with no algorithmic results combining the two. Our main result is an efficient algorithm for computing competitive equilibria in a setting encompassing both those generalizations. We assume that sellers have multiple copies of each items. A buyer i's quasi-linear utility is given by their gross substitute valuation for the bundle S of items they are assigned to, minus the sum of the payments qij(pj) for each item j ∈ S, where pj is the price of item j and qij is piecewise linear, strictly increasing. Our algorithm combines procedures for matroid intersection problems with augmenting forest techniques from matching theory. We also show that in a mild generalization of our model without quasilinear utilities, computing a competitive equilibrium is NP-hard. The full version of the paper can be found at https://arxiv.org/abs/2406.13620.

In dynamic submodular maximization, the goal is to maintain a high-value solution over a sequence of element insertions and deletions with a fast update time. Motivated by large-scale applications and the fact that dynamic data often exhibits patterns, we ask the following question: can predictions be used to accelerate the update time of dynamic submodular maximization algorithms? We consider the model for dynamic algorithms with predictions where predictions regarding the insertion and deletion times of elements can be used for preprocessing. Our main result is an algorithm with an $O(poly(\log \eta, \log w, \log k))$ amortized update time over the sequence of updates that achieves a $1/2 - \epsilon$ approximation in expectation for dynamic monotone submodular maximization under a cardinality constraint $k$, where the prediction error $\eta$ is the number of elements that are not inserted and deleted within $w$ time steps of their predicted insertion and deletion times. This amortized update time is independent of the length of the stream and instead depends on the prediction error.

In this work we introduce an alternative model for the design and analysis of strategyproof mechanisms that is motivated by the recent surge of work in "learning-augmented algorithms". Aiming to complement the traditional approach in computer science, which analyzes the performance of algorithms based on worst-case instances, this line of work has focused on the design and analysis of algorithms that are enhanced with machine-learned predictions regarding the optimal solution. The algorithms can use the predictions as a guide to inform their decisions, and the goal is to achieve much stronger performance guarantees when these predictions are accurate (consistency), while also maintaining near-optimal worst-case guarantees, even if these predictions are very inaccurate (robustness). So far, these results have been limited to algorithms, but in this work we argue that another fertile ground for this framework is in mechanism design. We initiate the design and analysis of strategyproof mechanisms that are augmented with predictions regarding the private information of the participating agents. To exhibit the important benefits of this approach, we revisit the canonical problem of facility location with strategic agents in the two-dimensional Euclidean space. We study both the egalitarian and utilitarian social cost functions, and we propose new strategyproof mechanisms that leverage predictions to guarantee an optimal trade-off between consistency and robustness guarantees. This provides the designer with a menu of mechanism options to choose from, depending on her confidence regarding the prediction accuracy. Furthermore, we also prove parameterized approximation results as a function of the prediction error, showing that our mechanisms perform well even when the predictions are not fully accurate.

In this work, we study the classical problem of maximizing a submodular function subject to a matroid constraint. We develop deterministic algorithms that are very parsimonious with respect to querying the submodular function, for both the case when the submodular function is monotone and the general submodular case. In particular, we present a 1/4 approximation algorithm for the monotone case that uses exactly one query per element, which gives the same total number of queries n as the number of queries required to compute the maximum singleton. For the general case, we present a constant factor approximation algorithm that requires 2 queries per element, which is the first algorithm for this problem with linear query complexity in the size of the ground set.

Algorithms with predictions is a recent framework for decision-making under uncertainty that leverages the power of machine-learned predictions without making any assumption about their quality. The goal in this framework is for algorithms to achieve an improved performance when the predictions are accurate while maintaining acceptable guarantees when the predictions are erroneous. A serious concern with algorithms that use predictions is that these predictions can be biased and, as a result, cause the algorithm to make decisions that are deemed unfair. We show that this concern manifests itself in the classical secretary problem in the learning-augmented setting---the state-of-the-art algorithm can have zero probability of accepting the best candidate, which we deem unfair, despite promising to accept a candidate whose expected value is at least $\max\{\Omega (1) , 1 - O(\varepsilon)\}$ times the optimal value, where $\varepsilon$ is the prediction error. We show how to preserve this promise while also guaranteeing to accept the best candidate with probability $\Omega(1)$. Our algorithm and analysis are based on a new ``pegging'' idea that diverges from existing works and simplifies/unifies some of their results. Finally, we extend to the $k$-secretary problem and complement our theoretical analysis with experiments.

We consider the online minimum cost matching problem on the line, in which there are n servers and, at each of n time steps, a request arrives and must be irrevocably matched to a server that has not yet been matched to, with the goal of minimizing the sum of the distances between the matched pairs. Despite achieving a worst-case competitive ratio that is exponential in n, the simple greedy algorithm, which matches each request to its nearest available free server, performs very well in practice. A major question is thus to explain greedy's strong empirical performance. In this paper, we aim to understand the performance of greedy over instances that are at least partially random. When both the requests and the servers are drawn uniformly and independently from [0,1], we show that greedy is constant competitive, which improves over the previously best-known O(√(n)) bound. We extend this constant competitive ratio to a setting with a linear excess of servers, which improves over the previously best-known O(log^3n) bound. We moreover show that in the semi-random model where the requests are still drawn uniformly and independently but where the servers are chosen adversarially, greedy achieves an O(logn) competitive ratio. When the requests arrive in a random order but are chosen adversarially, it was previously known that greedy is O(n)-competitive. Even though this one-sided randomness allows a large improvement in greedy's competitive ratio compared to the model where requests are adversarial and arrive in a random order, we show that it is not sufficient to obtain a constant competitive ratio by giving a tight Ω(logn) lower bound. These results invite further investigation about how much randomness is necessary and sufficient to obtain strong theoretical guarantees for the greedy algorithm for online minimum cost matching, on the line and beyond.

A surge of recent work has focused on analyzing the performance of algorithms guided by predictions, aiming to enhance their worst-case performance guarantees with improved guarantees when the predictions are accurate. This "learning-augmented" framework was recently also extended to mechanism design settings involving strategic agents and we provide an overview of these results.

Worst-case analysis is a performance measure that is often too pessimistic to indicate which algorithms we should use in practice. A classical example is in the context of the Euclidean Traveling Salesman Problem (TSP) in the plane, where local search performs very well in practice even though it only achieves an Ω (log n/loglog n) worst-case approximation ratio. In such cases, a natural alternative approach to worst-case analysis is to analyze the performance of algorithms in semi-random models. In this paper, we propose and investigate a novel semi-random model for the Euclidean TSP. In this model, called the simultaneous semi-random model, an instance over n points consists of the union of an adversarial instance over (1-α )n points and a random instance over α n points, for some α∈ [0, 1] . As with smoothed analysis, the semi-random model interpolates between distributional (random) analysis when α = 1 and worst-case analysis when α = 0 . In contrast to smoothed analysis, this model trades off allowing some completely random points in order to have other points that exhibit a fully arbitrary structure. We show that with only an α = 1/log n fraction of the points being random, local search achieves an 𝒪(loglog n) approximation in the simultaneous semi-random model for Euclidean TSP in fixed dimensions. On the other hand, we show that at least a polynomial number of random points are required to obtain an asymptotic improvement in the approximation ratio of local search compared to its worst-case approximation, even in two dimensions.

Algorithms with predictions is a recent framework that has been used to overcome pessimistic worst-case bounds in incomplete information settings. In the context of scheduling, very recent work has leveraged machine-learned predictions to design algorithms that achieve improved approximation ratios in settings where the processing times of the jobs are initially unknown. In this paper, we study the speed-robust scheduling problem where the speeds of the machines, instead of the processing times of the jobs, are unknown and augment this problem with predictions. Our main result is an algorithm that achieves a $\min\{\eta^2(1+\alpha), (2 + 2/\alpha)\}$ approximation, for any $\alpha \in (0,1)$, where $\eta \geq 1$ is the prediction error. When the predictions are accurate, this approximation outperforms the best known approximation for speed-robust scheduling without predictions of $2-1/m$, where $m$ is the number of machines, while simultaneously maintaining a worst-case approximation of $2 + 2/\alpha$ even when the predictions are arbitrarily wrong. In addition, we obtain improved approximations for three special cases: equal job sizes, infinitesimal job sizes, and binary machine speeds. We also complement our algorithmic results with lower bounds. Finally, we empirically evaluate our algorithm against existing algorithms for speed-robust scheduling.

In their seminal paper that initiated the field of algorithmic mechanism design, \citet{NR99} studied the problem of designing strategyproof mechanisms for scheduling jobs on unrelated machines aiming to minimize the makespan. They provided a strategyproof mechanism that achieves an $n$-approximation and they made the bold conjecture that this is the best approximation achievable by any deterministic strategyproof scheduling mechanism. After more than two decades and several efforts, $n$ remains the best known approximation and very recent work by \citet{CKK21} has been able to prove an $\Omega(\sqrt{n})$ approximation lower bound for all deterministic strategyproof mechanisms. This strong negative result, however, heavily depends on the fact that the performance of these mechanisms is evaluated using worst-case analysis. To overcome such overly pessimistic, and often uninformative, worst-case bounds, a surge of recent work has focused on the ``learning-augmented framework'', whose goal is to leverage machine-learned predictions to obtain improved approximations when these predictions are accurate (consistency), while also achieving near-optimal worst-case approximations even when the predictions are arbitrarily wrong (robustness). In this work, we study the classic strategic scheduling problem of~\citet{NR99} using the learning-augmented framework and give a deterministic polynomial-time strategyproof mechanism that is $6$-consistent and $2n$-robust. We thus achieve the ``best of both worlds'': an $O(1)$ consistency and an $O(n)$ robustness that asymptotically matches the best-known approximation. We then extend this result to provide more general worst-case approximation guarantees as a function of the prediction error. Finally, we complement our positive results by showing that any $1$-consistent deterministic strategyproof mechanism has unbounded robustness.

We address an energy-efficient scheduling problem for practical multiple-input single-output (MISO) systems with stringent execution-time requirements. Optimal user-group scheduling is adopted to enable timely and energy-efficient data transmission, such that all the users' demand can be delivered within a limited time. The high computational complexity in optimal iterative algorithms limits their applications in real-time network operations. In this paper, we rethink the conventional optimization algorithms, and embed machine-learning based predictions in the optimization process, aiming at improving the computational efficiency and meeting the stringent execution-time limits in practice, while retaining competitive energy-saving performance for the MISO system. Numerical results demonstrate that the proposed method, i.e., optimization with machine-learning predictions (OMLP), is able to provide a time-efficient and high-quality solution for the considered scheduling problem. Towards online scheduling in real-time communications, OMLP is of high computational efficiency compared to conventional optimal iterative algorithms. OMLP guarantees the optimality as long as the machine-learning based predictions are accurate.